#7648766

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(10,13)(14,16)(18,21)(19,23);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18,20,21)(19,22,23,24);(2,5)(6,8)(17,18)(19,22)(20,21)(23,24);(3,7)(6,8)(17,19)(18,22)(20,23)(21,24);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16);(1,2)(3,6)(4,5)(7,8)

Polycyclic presentation:

2^9|b^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555962);